Integer right triangles
Pythagorean Triples
A Pythagorean triple is a set of three positive integers (a, b, c) where
a² + b² = c². This page covers the definition, Euclid's generation formula, a list
of 20 common triples, and patterns to help you recognize them instantly.
Definition
What Are Pythagorean Triples?
A Pythagorean triple is a set of three positive integers (a, b, c) such that
a² + b² = c². These integers represent the side lengths of a right triangle where
all three sides are whole numbers. The most famous example is (3, 4, 5):
3² + 4² = 9 + 16 = 25 = 5².
If you want a refresher on the theorem itself before working with integer triples, see The Pythagorean Theorem.
Primitive Pythagorean Triple
Definition: gcd(a, b, c) = 1. The three numbers share no common
factor other than 1.
Examples: (3, 4, 5), (5, 12, 13), (8, 15, 17).
Feature: A primitive triple cannot be reduced any further.
Non-Primitive Pythagorean Triple
Definition: gcd(a, b, c) > 1. The three numbers have a common
factor.
Examples: (6, 8, 10), (9, 12, 15), (10, 24, 26).
Feature: Every non-primitive triple is a multiple of a primitive triple.
For worked side-finding problems, see Pythagorean Theorem Examples. For checking whether three side lengths form a right triangle, see Converse of the Pythagorean Theorem.
Generation formula
Euclid's Formula for Generating Pythagorean Triples
Euclid's formula generates all primitive Pythagorean triples. Choose two positive integers
m and n such that m > n, gcd(m, n) = 1,
and m and n have opposite parity, one even and one odd.
a = m² − n²
b = 2mn
c = m² + n²
| m | n | a = m² − n² | b = 2mn | c = m² + n² | Triple |
|---|---|---|---|---|---|
| 2 | 1 | 3 | 4 | 5 | (3, 4, 5) |
| 3 | 2 | 5 | 12 | 13 | (5, 12, 13) |
| 4 | 1 | 15 | 8 | 17 | (8, 15, 17) |
| 4 | 3 | 7 | 24 | 25 | (7, 24, 25) |
| 5 | 2 | 21 | 20 | 29 | (20, 21, 29) |
| 5 | 4 | 9 | 40 | 41 | (9, 40, 41) |
Note: Euclid's formula with these conditions generates all
primitive triples. To get non-primitive triples as well, multiply any primitive
triple by a positive integer k: (ka, kb, kc).
To verify any triple, simply check a² + b² = c². Example:
(3, 4, 5) → 9 + 16 = 25 = 5² ✓
Reference list
List of Common Pythagorean Triples
These 20 triples are the ones students and teachers encounter most often. The left table shows primitive triples. The right table shows common non-primitive multiples.
Primitive Pythagorean Triples
These are the first 10 high-frequency primitive triples, each already in lowest terms.
| # | a | b | c | Verify |
|---|---|---|---|---|
| 1 | 3 | 4 | 5 | 9 + 16 = 25 ✓ |
| 2 | 5 | 12 | 13 | 25 + 144 = 169 ✓ |
| 3 | 8 | 15 | 17 | 64 + 225 = 289 ✓ |
| 4 | 7 | 24 | 25 | 49 + 576 = 625 ✓ |
| 5 | 20 | 21 | 29 | 400 + 441 = 841 ✓ |
| 6 | 9 | 40 | 41 | 81 + 1600 = 1681 ✓ |
| 7 | 12 | 35 | 37 | 144 + 1225 = 1369 ✓ |
| 8 | 11 | 60 | 61 | 121 + 3600 = 3721 ✓ |
| 9 | 28 | 45 | 53 | 784 + 2025 = 2809 ✓ |
| 10 | 33 | 56 | 65 | 1089 + 3136 = 4225 ✓ |
Non-Primitive Pythagorean Triples
These are common scaled versions of primitive triples. Each row shows the source triple.
| # | a | b | c | Source (×k of) |
|---|---|---|---|---|
| 1 | 6 | 8 | 10 | 2 × (3, 4, 5) |
| 2 | 9 | 12 | 15 | 3 × (3, 4, 5) |
| 3 | 12 | 16 | 20 | 4 × (3, 4, 5) |
| 4 | 15 | 20 | 25 | 5 × (3, 4, 5) |
| 5 | 10 | 24 | 26 | 2 × (5, 12, 13) |
| 6 | 16 | 30 | 34 | 2 × (8, 15, 17) |
| 7 | 20 | 48 | 52 | 4 × (5, 12, 13) |
| 8 | 14 | 48 | 50 | 2 × (7, 24, 25) |
| 9 | 24 | 45 | 51 | 3 × (8, 15, 17) |
| 10 | 21 | 72 | 75 | 3 × (7, 24, 25) |
This list is only a starting point. There are infinitely many Pythagorean triples, and Euclid's formula gives a systematic way to generate them.
Recognition tips
Patterns and Recognition Tips
You do not need to memorize hundreds of triples. A few number patterns let you spot the most common ones almost instantly.
The Odd Number Pattern
For any odd number a ≥ 3, there is a Pythagorean triple built directly from
a.
a, (a² − 1) / 2, (a² + 1) / 2
Examples: (3, 4, 5), (5, 12, 13), (7, 24, 25), (9, 40, 41).
The Even Number Pattern
For any even number a ≥ 4, let n = a / 2. Then one Pythagorean
triple is:
a, n² − 1, n² + 1
Examples: (3, 4, 5), (6, 8, 10), (8, 15, 17), (10, 24, 26).
Multiples Stay Triples
If (a, b, c) is a Pythagorean triple, then multiplying every term by the same
positive integer k produces another triple.
(ka, kb, kc)
Examples: 2 × (3, 4, 5) = (6, 8, 10), 3 × (5, 12, 13) = (15, 36, 39).
Primitive Triples Have One Even Leg
In every primitive triple, one leg is even, the other leg is odd, and the hypotenuse is odd. This is a quick sanity check when you are looking at a candidate triple.
even leg + odd leg → odd hypotenuse
Examples: (3, 4, 5), (5, 12, 13), (20, 21, 29).
The Consecutive Large-Number Family
In the odd-number pattern, the two larger numbers differ by exactly 1. That is
why so many famous triples end with consecutive values.
(a² + 1) / 2 − (a² − 1) / 2 = 1
Examples: (3, 4, 5), (5, 12, 13), (7, 24, 25), (9, 40, 41).
FAQ
Frequently Asked Questions
Pythagorean triples are sets of three positive integers (a, b, c) that satisfy
a² + b² = c². The most familiar example is (3, 4, 5), because
9 + 16 = 25.
A standard method is Euclid's formula: choose integers m > n > 0, then
compute a = m² − n², b = 2mn, and c = m² + n². If
m and n are coprime and have opposite parity, the result is a
primitive triple.
A primitive triple is one where the three numbers share no common factor other than
1. For example, (3, 4, 5) is primitive, but
(6, 8, 10) is not because all three numbers are divisible by 2.
Compute the greatest common divisor of all three numbers. If
gcd(a, b, c) = 1, the triple is primitive. If the gcd is larger than
1, the triple is a scaled multiple of a smaller triple.
Yes. There are infinitely many primitive triples and therefore infinitely many non-primitive
triples as well. Euclid's formula already produces an endless stream of new primitive
examples as m and n grow.
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